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Traditional computational methods for studying quantum many-body systems are
"forward methods," which take quantum models, i.e., Hamiltonians, as input and
produce ground states as output. However, such forward methods often limit
one's perspective to a small fraction of the space of possible Hamiltonians. We
introduce an alternative computational "inverse method," the
Eigenstate-to-Hamiltonian Construction (EHC), that allows us to better
understand the vast space of quantum models describing strongly correlated
systems. EHC takes as input a wave function $|\psi_T\rangle$ and produces as
output Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This is
accomplished by computing the quantum covariance matrix, a quantum mechanical
generalization of a classical covariance matrix. EHC is widely applicable to a
number of models and in this work we consider seven different examples. Using
the EHC method, we construct a parent Hamiltonian with a new type of triplet
dimer ground state, a parent Hamiltonian with two different targeted degenerate
ground states, and large classes of parent Hamiltonians with the same ground
states as well-known quantum models, such as the Majumdar-Ghosh model, the XX
chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model. EHC gives an
alternative inverse approach for studying quantum many-body phenomena.